If you want the light strings to be evenly spaced, then the tree will not work as a post. However, there must be something to anchor them to at the end of the patio. I suggest you build a simple arbor of two posts connected with beams that has smaller lumber that sits on top of the beams. This is an architectural statement feature that provides a place to attach your lights. The one below is quite massive, but your does not need to be.

Pergolas and Arbors · More Info

I ran across this SITE that explains how to build this style of arbor. Theirs is intended to hold a swing, but the build is pretty much the same. You can leave your posts taller than they did. The posts just need to be the same height so the beams sit level across the span.

I’d go from end of house to inside corner (along gutter) up to tree (higher point by several feet) and then back to corner.
You could do just a triangle of lights. Or use a rope for the triangle then just zig-zag lights between. Or do both for maximum lights. Coral = rope or lights; blue = zigzagged lights in edited pics below.

Adalisa- that zig zag design is very pretty. I had not thought of that. Thank you!

Mindshidt- where would you place the arbor?

I would put it on the end opposite the house. However, string lights are most often in lengths of 25, 50, or 100 feet, usually with extra wire to the power source. So it would be best to make sure the distance from the house fascia to an arbor is within the available string length. Here are several internet sellers Link1, Link2, Link3. And, of course, you can find lights via Amazon, Wayfair, Lowes, Home Depot and Target.

While I was looking up the above links I happened on these two sites that use large planters as a base to set posts. It is a so-easy idea. LinkA, LinkB The posts in planters are not as stable as posts set in the ground, but you could set the planters where they work the best for you. Hope this helps.